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Sharp Isoperimetric Inequalities

for Affine Quermassintegrals

Emanuel Milman∗,† Amir Yehudayoff∗,‡

Abstract

The affine quermassintegrals associated to a convex body inRn are affine-invariant

analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, andthus constitute a central pillar of affine convex geometry. They were introducedin the 1980’s by E. Lutwak, who conjectured that among all convex bodies of agiven volume, the k-th affine quermassintegral is minimized precisely on the familyof ellipsoids. The known cases k = 1 and k = n − 1 correspond to the classicalBlaschke–Santalo and Petty projection inequalities, respectively. In this work weconfirm Lutwak’s conjecture, including characterization of the equality cases, for allvalues of k = 1, . . . , n − 1, in a single unified framework. In fact, it turns out thatellipsoids are the only local minimizers with respect to the Hausdorff topology.

In addition, we address a related conjecture of Lutwak on the validity of certainAlexandrov–Fenchel-type inequalities for affine (and more generally Lp-moment)quermassintegrals. The case p = 0 corresponds to a sharp averaged Loomis–Whitneyisoperimetric inequality. Finally, a new extremely simple proof of Petty’s projectioninequality is presented, revealing a certain duality relation with the Blaschke–Santaloinequality.

∗Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel.†Email: emilman@tx.technion.ac.il.‡Email: amir.yehudayoff@gmail.com.2010 Mathematics Subject Classification: 52A40.Keywords: Lutwak’s Conjecture, Affine Quermassintegrals, Convex Geometry, Isoperimetric Inequal-

ity, Blaschke–Santalo inequality, Petty’s projection inequality, Loomis–Whitney inequality.The research leading to these results is part of a project that has received funding from the European

Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 637851) and from the Israeli Science Foundation grant #1162/15.

1

1 Introduction

Let K denote a convex compact set with non-empty interior (“convex body”) in Eu-clidean space R

n. Given k = 1, . . . , n, the k-th affine quermassintegral of K was definedby E. Lutwak in [33] as:

Φk(K) :=|Bn

2 ||Bk

2 |

(

∫

Gn,k

|PFK|−nσ(dF )

)− 1n

. (1.1)

Here, Bn2 denotes the Euclidean unit ball in R

n, Gn,k denotes the Grassmannian of allk-dimensional linear subspaces of Rn endowed with its unique Haar probability measureσ, PF denotes orthogonal projection onto F ∈ Gn,k, and | · | denotes Lebesgue measureon the corresponding linear space. Note our convention of using the index k instead ofthe more traditional n − k above, as this reflects the order of homogeneity of Φk(K)under scaling of K. It was shown by Grinberg [26] that K 7→ Φk(K) is indeed invariantunder volume-preserving affine transformations (or simply “affine-invariant”).

One of the most important problems in higher-rank affine convex geometry is toobtain sharp lower and upper bounds on Φk(K) and to characterize the extremizers.The only sharp results obtained thus far have been for the rank-one classical cases k = 1and k = n− 1 (see below). In this work, we establish the following theorem, conjecturedby Lutwak in [35] (see also [36, Open Problem 12.3], [23, Problem 9.3], [57, (9.57)]):

Theorem 1.1. For any convex body K ⊂ Rn and k = 1, . . . , n− 1:

Φk(K) ≥ Φk(BK), (1.2)

with equality for a given k if and only if K is an ellipsoid.

Here and throughout this work, BK denotes the (centered) Euclidean ball having thesame volume as K.

The cases k = 1 and k = n− 1 above are completely classical. For origin-symmetricconvex bodies, Φ−n

1 (K) is proportional to the volume of the polar body K◦, and so thecase k = 1 of (1.2) amounts to the Blaschke–Santalo inequality:

|K||K◦| ≤ |Bn2 |2. (1.3)

For general convex bodies, it is easy to check that (1.2) for k = 1 is weaker than theBlaschke–Santalo inequality, stating that (1.3) holds when K is first centered at itsSantalo point – in that case, the corresponding polar body is denoted by K◦,s. TheBlaschke–Santalo inequality was established by Blaschke [6] for n ≤ 3 and Santalo [56]for general n by using the isoperimetric inequality for affine surface area. The char-acterization of ellipsoids as the only cases of equality was established by Blaschke andSantalo under certain regularity assumptions on K, which were removed when K = −Kby Saint-Raymond [55], who also gave a simplified proof of the origin-symmetric case.

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For general convex bodies without any regularity assumptions, the equality conditionsof the Blaschke–Santalo inequality (and hence of (1.2) for k = 1) were established byPetty [52]. Subsequent additional proofs (most of which include a characterization ofequality) were obtained by Meyer–Pajor [44], Lutwak–Zhang [40], Lutwak–Yang–Zhang[38], Campi–Gronchi [16] and Meyer–Reisner [45], to name a few.

On the other extreme, Φ−nn−1(K) is proportional to the volume of the polar projection

body Π∗K, and so the case k = n − 1 of (1.2) amounts to Petty’s projection inequality[50]:

|Π∗K| ≤ |Π∗BK |,with equality if and only ifK is an ellipsoid. Petty derived this result from the Busemann–Petty centroid inequality (see [23, Chapter 9]), and the two inequalities are in factequivalent to each other [34, 36]. Subsequent additional proofs of Petty’s projectioninequality with characterization of equality and extensions thereof were obtained byLutwak–Yang–Zhang [37, 39] and Campi–Gronchi [15] (in fact, for the more general Lp

and Orlicz projection bodies). In [62], Zhang extended Petty’s projection inequality tomore general compact sets, and showed that this more general version is equivalent to asharp affine-invariant Sobolev inequality which is stronger than the classical (non-affine)one (and hence stronger than the classical isoperimetric inequality).

These classical cases k ∈ {1, n − 1} are fundamental tools in Affine Convex Geome-try, and have found further applications in Asymptotic Geometric Analysis, FunctionalInequalities and Concentration of Measure, Partial Differential Equations, FunctionalAnalysis, the Geometry of Numbers, Discrete Geometry and Polytopal Approximations,Stereology and Stochastic Geometry, and Minkowskian Geometry (see [36, 2] and thereferences therein).

The remaining cases k = 2, . . . , n− 2 of Theorem 1.1 are entirely new and constitutethe main result of this work.

Let us briefly mention some additional related results. By employing methods fromAsymptotic Geometric Analysis, Paouris–Pivovarov [49] (see also [19, 20]) have previ-ously confirmed the inequality (1.2) up to a factor of ck for some constant c > 0. Zouand Xiong [63] have shown that Φk(K) is lower-bounded (up to normalization) by thevolume of the k-th projection mean ellipsoid of K, which is however incomparable to thevolume of K. Lutwak also proposed the related notion of dual affine quermassintegralsΦk(K), in which one replaces projections by sections and the L−n-norm by the L+n onein the definition of Φk. The analogous problem of obtaining a sharp upper bound onΦk(K) for k = 2, . . . , n − 1 was resolved for convex bodies by Busemann–Straus [14,p. 70] and Grinberg [26], who also characterized centered ellipsoids as the only cases ofequality (see also [58, Section 8.6]). These results for Φk(K) were extended to arbitrarybounded Borel sets K (for which the characterization of equality is much more delicate)by Gardner [24]. Finally, the question of obtaining sharp upper bounds on Φk(K) hasa long history and deserves a survey in itself. Let us only mention that a sharp up-per bound on Φ1(K) amounts to Mahler’s conjecture [42] (see also [36, Section 12.1]),

3

stating that the volume product |K||K◦,s| for general convex bodies K is maximized onsimplices, and on cubes for origin-symmetric K. This has been confirmed by Mahler [41]in R

2, by Iriyeh–Shibata [30] (see also [21]) for origin-symmetric K in R3, and up to a

factor of cn by Bourgain and V. Milman [10] (see also [31, 47, 25]). A sharp upper boundon Φn−1(K) (i.e. reverse Petty projection inequality) with characterization of simplicesas the only cases of equality was obtained by Zhang [61]. To the best of our knowledge,sharp upper bounds on Φk(K) for 1 ≤ k ≤ n − 2 and n ≥ 4 remain wide open; someasymptotic non-sharp estimates may be found in [19, 49, 20], see also [17]. We refer tothe excellent monographs [23, 57, 58] and survey paper [36] for additional exposition andcontext.

1.1 Steiner symmetrization

Not surprisingly, our proof of Theorem 1.1 proceeds by using the classical tool of Steinersymmetrization. Let SuK denote the Steiner symmetral of K in a given direction u ∈Sn−1 – we refer to Section 2 for missing standard definitions. We obtain the following

stronger version of both the inequality and equality case announced in Theorem 1.1:

Theorem 1.2. For any convex body K ⊂ Rn, k = 1, . . . , n− 1 and u ∈ S

n−1:

Φk(K) ≥ Φk(SuK), (1.4)

with equality for a given k and all u ∈ Sn−1 if and only if K is an ellipsoid.

The inequality (1.4) for k = 1 when K = −K is origin symmetric was obtained byMeyer–Pajor [44] in their proof of the Blaschke–Santalo inequality (see also Lutwak–Zhang [40] and Campi–Gronchi [16]). For k = n− 1, (1.4) was shown by Lutwak–Yang–Zhang [37, 39] in their proof of Petty’s projection inequality (for the more general Lp

and Orlicz projection bodies). The cases k = 2, . . . , n − 2 of (1.4) are new.

Surprisingly, the equality case of Theorem 1.2 was, to the best of our knowledge,previously only known in the case k = 1: for origin-symmetric convex K = −K thisis due to Meyer–Pajor [44] (see also Lutwak–Zhang [40]); Meyer–Reisner [45] prove ananalogous result for the Blaschke–Santalo inequality for general convex bodies. Even inthe classical case k = n − 1 corresponding to Petty’s projection inequality, the equalitycase of Theorem 1.2 appears to be new; note that for Lp-projection inequalities with1 < p < ∞ and more general strictly convex Orlicz functions, an analogous result wasobtained by Lutwak–Yang–Zhang [37, 39], but their equality analysis breaks down in theclassical case p = 1. This is consistent with our own analysis in this work, where thecase of equality when 1 ≤ k ≤ n− 2 is relatively simpler, but the case k = n− 1 involvesa fair amount of additional work. It is worthwhile to note that our approach avoids anyregularity issues in both the proof of the inequality and in the analysis of the equalitycases, in contrast to some other approaches in the classical cases k = 1 and k = n− 1.

A different (yet very related) strengthening of Theorem 1.1 is given by:

Theorem 1.3. For all k = 1, . . . , n−1, among all convex bodies in Rn of a given volume,

ellipsoids are the only local minimizers of Φk with respect to the Hausdorff topology.

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For k = 1 this was recently shown by Meyer–Reisner [46] (in fact, they show thatan analogous statement holds for the volume of K◦,s, yielding a slightly stronger resultthan the one above in the case of non-origin-symmetric convex bodies). The cases k =2, . . . , n− 2 including the classical case k = n− 1 for the volume of the polar projectionbody are new.

1.2 The challenge

All proofs of the classical cases k ∈ {1, n− 1} commence by associating to K a (convex)body Lk(K) in R

n which encodes the function Gn,k ∋ F 7→ |PFK|. In these cases, thisis easy to do: by identifying Gn,k with RP

n−1, extending the function homogeneouslyto R

n, and considering its level set, one obtains (up to normalization) the polar body(k = 1) and polar projection body (k = n − 1) of K. In particular, the volume ofLk(K) coincides with Φ−n

k (K), and the fact that Lk(K) resides in a linear space makesit convenient for checking the effect of Steiner symmetrization of K on |Lk(K)|. Forother values of k, it is not at all clear what is the right body Lk(K) to associate withthe function Gn,k ∋ F 7→ |PFK|, and more importantly, in which space it should reside,as the standard ways of mapping a linear space (such as (Rn)k) onto the cone over Gn,k

are highly non-injective.

Our proof utilizes a new body which we call the Projection Rolodex of K. It doesnot reside in a linear space, but rather (as perhaps its name suggests) in a vector bundleover a lower-dimensional Grassmannian. Another difference with the classical cases,where the body Lk(K) depends on K alone, is that the Rolodex Lk,u(K) also dependson the direction u ∈ S

n−1 in which we perform the Steiner symmetrization. The pricewe pay is that it is not the usual Haar measure of Lk,u(K) which is related to Φ−n

k (K),but rather some auxiliary measure µu which we introduce. We thus replace the order ofquantifiers compared to the classical proofs: we first select a direction u, only then definethe Rolodex Lk,u(K), and now our task is to verify that µu(Lk,u(K)) ≤ µu(Lk,u(SuK)).The remaining challenge is then to analyze how Steiner symmetrization affects |PFK|for F ∈ Gn,k, and so we embark on a systematic study of the latter in Section 3 (in fact,for general shadow systems).

The above scheme allows us to prove Theorems 1.1, 1.2 and 1.3 simultaneously for allvalues of k in a single unified framework, revealing a surprising connection between theBlaschke–Santalo inequality and Petty’s projection inequality. From this point of view,Petty’s inequality may be interpreted as an integrated form of a generalized Blaschke–Santalo inequality for a new family of polar-bodies associated with a given convex bodyK, encoded by the Projection Rolodex. We do not know whether the Blaschke–Santaloinequality may dually be interpreted as a generalized Petty projection inequality. How-ever, in Subsection 8.2 we obtain a new extremely simple proof of Petty’s projectioninequality, which reveals a deeper duality with the Blaschke–Santalo inequality.

5

1.3 Lp-moment quermassintegrals and averaged Loomis–Whitney

An analogous statement to that of Theorem 1.1 holds for the Lp-moment quermassinte-grals Qk,p, replacing the L−n-norm by the Lp-norm in the definition (1.1):

Qk,p(K) ≥ Qk,p(BK) ∀p ≥ −n, (1.5)

with equality for p > −n if and only if K is a Euclidean ball – see Definition 7.1 andTheorem 7.2. For p = 1 these are nothing but the classical isoperimetric inequalitiesfor the intrinsic volumes Wk(K), and for p = −1 the corresponding isoperimetric in-equalities for the harmonic quermassintegrals Wk(K) were established by Lutwak [35]by bootstrapping Petty’s projection inequality. It is possible to extend this bootstrap-ping all the way down to the value p = −(k + 1), see Remark 7.7. Of course, Jensen’sinequality implies that the family of inequalities (1.5) becomes stronger as p decreases,and so our result for p = −n is stronger than all of the above. It is easy to check that thevalue p = −n is best possible, i.e. that (1.5) is simply false for p < −n, see Remark 7.3.Going below p = −(k+1) all the way down to the optimal value p = −n requires severalnew ideas when 2 ≤ k ≤ n− 2, as outlined above.

It is worthwhile to note that the case p = 0 is of special interest, as (1.5) maythen be interpreted as a sharp averaged Loomis–Whitney isoperimetric inequality. Theclassical Loomis–Whitney inequality [32] lower bounds the geometric average of all k-dimensional projections of a compact setK onto the principle axes in terms of the volumeof K, yielding a sharp result for the cube (aligned with the axes). As an application,Loomis and Whitney deduce an isoperimetric inequality for the surface area of K, butwith non-sharp constant. This is expected, as their inequality depends on the choice ofcoordinate system. The case p = 0 of (1.5) implies that if one chooses the coordinatesystem at random and takes the geometric average of all k-dimensional projections, animprovement over the original Loomis–Whitney inequality is possible (for convex K).Moreover, this improvement is sharp for the Euclidean ball and thus yields the sharpconstant in the classical isoperimetric inequality for the surface area – see Subsection 7.2.

1.4 Alexandrov–Fenchel-type inequalities

It is convenient to introduce:

Ik,p(K) :=Qk,p(K)

Qk,p(BK)=

(∫

Gn,k|PFK|pσ(dF )

∫

Gn,k|PFBK |pσ(dF )

)1p

.

Note that Ik,p(B) = 1 for any Euclidean ball B and all k, p, that In,p(K) = 1 for all p,and that (1.5) translates to Ik,p(K) ≥ 1 for all p ≥ −n.

In the classical case p = 1, Alexandrov’s inequalities [57, 22] (a particular case of theAlexandrov–Fenchel inequalities) assert that:

I1,1(K) ≥ I1/22,1 (K) ≥ . . . ≥ I1/k

k,1 (K) ≥ . . . ≥ I1/(n−1)n−1,1 (K) ≥ I1/n

n,1 (K) = 1.

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The following was proved by Lutwak for p = −1 and conjectured to hold for p = −n in[35] (see also [23, Problem 9.5]):

Conjecture 1.4. For all p ∈ [−n, 0] and for any convex body K in Rn:

I1,p(K) ≥ I1/22,p (K) ≥ . . . ≥ I1/k

k,p (K) ≥ . . . ≥ I1/(n−1)n−1,p (K) ≥ I1/n

n,p (K) = 1.

Our isoperimetric inequality (1.5) establishes the inequality between each of the termsand the last one. In the next theorem, which builds upon Theorem 1.1, we confirm “half”of the above conjecture.

Theorem 1.5. For every p ∈ [−n, 0] and 1 ≤ k ≤ m ≤ n:

I1/kk,p (K) ≥ I1/m

m,p (K),

for any convex body K in Rn whenever m ≥ −p. When k < m < n, equality holds for

p ≥ −m if and only if K is a Euclidean ball. When k < m = n, equality holds forp > −n (p = −n) if and only if K is a Euclidean ball (ellipsoid).

In particular, this confirms the conjecture for all p ∈ [−2, 0], recovering the casep = −1 established by Lutwak in [35]. The analogous statement for the dual Lp-momentquermassintegrals Ik,p for arbitrary bounded Borel sets (replacing p by −p and with thedirection of the inequality reversed) was established by Gardner [24, Theorem 7.4] inexactly the same corresponding range of parameters – see Subsection 7.3. EstablishingConjecture 1.4 in the remaining half range 1 ≤ k < m < −p is a fascinating problem.Lutwak’s original conjecture (the case p = −n above) is presently only established in theplane and for m = n (and all k) by Theorem 1.5.

Organization

The rest of this work is organized as follows. In Section 2 we introduce some standardnotation. In Section 3 we provide a proof of the sharp inequalities (1.2) and (1.4) ofTheorems 1.1 and 1.2. In Section 4 we establish some convexity properties which we willneed for the proof of Theorem 1.3. In Section 5 we provide a proof of the equality casesof Theorems 1.1 and 1.2 as well as Theorem 1.3 in the range 1 ≤ k ≤ n − 2; the casek = n−1 is treated in Section 6. In Section 7 we study the Lp-moment quermassintegralsQk,p(K) and establish Theorem 1.5; an interesting interpretation of the case p = 0 asa sharp averaged Loomis–Whitney isoperimetric inequality is described in Subsection7.2. In Section 8 we provide some concluding remarks – in Subsection 8.1 we discusspossible extensions of Theorem 1.1 to more general compact sets, and in Subsection 8.2we present a new simple proof of Petty’s projection inequality.

Acknowledgments. We thank Richard Gardner, Erwin Lutwak, Rolf Schneider andGaoyong Zhang for their comments on a preliminary version of this manuscript.

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2 Notation

For a real number a ∈ R, denote a+ := max(a, 0) and a− := (−a)+ so that a = a+ − a−.Denote R+ := [0,∞) and R− := (−∞, 0].

Given a Euclidean space E, we denote by BE its Euclidean unit ball and by S(E) =∂BE the corresponding unit-sphere; when E = (Rn, 〈·, ·〉) we write Bn

2 and Sn−1, respec-

tively. We write |x| for the Euclidean norm√

〈x, x〉. We denote by GE,k the Grassman-nian of all k-dimensional linear subspaces of E; when E = R

n, we simply write Gn,k.It is equipped with its SO(E)-invariant Haar probability measure, which we denote byσE,k, or simply σ when there is no risk of confusion. Here SO(E) denotes the groupof rotations on E, equipped with its invariant Haar probability measure σSO(E); whenE = R

n, we simply write SO(n). For a linear map T on E we write T ∗ for its adjointand T−∗ for the adjoint of its inverse if T is invertible.

We use LE to denote the Lebesgue measure on a k-dimensional affine subspace E;when the latter is clear from the context, we will simply write Lk. Recall that PE denotesorthogonal projection onto E. Given a compact set K in R

n, we use |K| and |PEK| asshorthand for Ln(K) and Lk(PEK), respectively. The Steiner symmetral of a compact

set K ⊂ Rn in the direction of u ∈ S

n−1, denoted SuK, is defined by requiring that theone-dimensional fiber SuK ∩ (y+ u⊥) is a symmetric interval about u⊥ having the sameone-dimensional Lebesgue measure as K ∩ (y + u⊥), for each y ∈ u⊥ so that the latteris non-empty. Clearly |SuK| = |K|, and it is well known that Steiner symmetrizationpreserves compactness as well as convexity [23, Chapter 2], [27, Chapter 9]. We denoteby Ru the reflection map about u⊥.

The support function hK and polar body K◦ of a compact set K ⊂ Rn are defined

as:hK(x) := max

y∈K〈x, y〉 , K◦ := {x ∈ R

n ; hK(x) ≤ 1}.

When in addition K is convex and contains the origin in its interior, we define:

‖x‖K := inf{t > 0 ; x ∈ tK},

so that K is precisely the unit ball of ‖·‖K . Note that in that case ‖x‖K = hK◦(x) andthat (K◦)◦ = K [57, Theorem 1.6.1].

While we will not require this for the sequel, we recall for completeness several notionsmentioned in the Introduction. The projection body ΠK of a convex bodyK, introduced(and shown to exist) by Minkowski, is defined as the convex body whose support functionsatisfies:

hΠK(θ) = |Pθ⊥K| ∀θ ∈ Sn−1.

The polar projection body Π∗K is defined as (ΠK)◦. The Santalo point s(K) of K isdefined as the unique point s in the interior of K for which |(K− s)◦| is minimized; K◦,s

is then defined as s(K) + (K − s(K))◦. We refer to [36] and the references therein forfurther details and context.

8

The Minkowski sum of two compact sets A,B ⊂ Rn is defined as A+B := {a+b ; a ∈

A, b ∈ B}. It is immediate to see that hA+B = hA+hB . The Hausdorff distance betweentwo compact subsets A,B of Rn is defined as the minimal ǫ > 0 so that A ⊂ B + ǫBn

2

and B ⊂ A+ ǫBn2 .

The classical Brunn–Minkowski inequality [57, 22, 23, 27] states that if K,L are twoconvex bodies in R

n then|K + L| 1n ≥ |K| 1n + |L| 1n ,

with equality if and only if L = λK + v for some λ > 0 and v ∈ Rn. An equivalent form

is given by Brunn’s concavity principle [27, Theorem 8.4], stating that if K is a convexbody in R

n+1 and u ∈ Sn then:

R ∋ t 7→ |K ∩ (tu+ u⊥)| 1n is concave on its support.

Finally, we say that K has a point of symmetry (at v ∈ Rn) if K − v = −(K − v).

3 Proof of the isoperimetric inequality

The inequality of Theorem 1.1 is a standard consequence of the following symmetrizationresult, already stated as (1.4) in Theorem 1.2:

Theorem 3.1. Steiner symmetrization in a direction u ∈ Sn−1 does not increase the

k-th affine quermassintegral Φk(K) for any convex body K ⊂ Rn and k = 1, . . . , n− 1:

Φk(K) ≥ Φk(SuK).

3.1 Ingredients

For the proof, we will need three main ingredients:

3.1.1 The Projection Rolodex

Fix k = 1, . . . , n. Given E ∈ Gn,k−1, x ∈ Rn and a compact set K ⊂ R

n, denote:

|PE∧xK| := |PE⊥x|Lk(Pspan(E,x)K). (3.1)

We will mainly consider the case when x ∈ E⊥, so that |PE⊥x| = |x|. By |PxK| we willmean |x||Pspan(x)K|, corresponding to the case E = {0} above.

We introduce the following two definitions, which may be of independent interest:

Definition 3.2. Given a compact set K ⊂ Rn and E ∈ Gn,k−1, the set

LE(K) := {x ∈ E⊥ ; |PE∧xK| ≤ 1} ⊂ E⊥

is called the E-projected polar body of K.

9

Note that LE(K) is always origin symmetric. An interesting property of LE(K) isthat it is always convex whenever K is (see Subsection 3.3). Indeed, this is immediateto see when E = {0} and K is convex, in which case we have:

L(K) := L{0}(K) = {x ∈ Rn ; hK(x) + hK(−x) ≤ 1} = (K −K)◦.

Hence, when K is an origin-symmetric convex body, L(K) coincides with 12K

◦. Contraryto the usual definition of polar body of a convex set, note that the above definition isinvariant under translations of K.

Definition 3.3. Given a compact set K ⊂ Rn, the set:

Lk,u(K) := {(E, xk) ; E ∈ Gu⊥,k−1, xk ∈ LE(K)}

is called the k-dimensional Projection Rolodex of K relative to u⊥.

The idea behind this definition is that it encodes the values of |PFK| for almostevery F ∈ Gn,k; indeed, we may write almost every F ∈ Gn,k as the direct sum ofE = F ∩ u⊥ ∈ Gu⊥,k−1 and span(θ) for θ ∈ S(E⊥), and use that tθ ∈ LE(K) iff|t| ≤ 1/|PFK| (so that |PFK| coincides with ‖θ‖LE(K)).

3.1.2 Convexity of shadow system’s projections

Our second main ingredient is the following key proposition, which pertains to a certainconvexity property of projections of shadow systems. We refer to Shephard [59] for ageneral treatise of shadow systems (introduced in an equivalent form by Rogers–Shephardin [54]), and only describe here what we need to formulate our claim.

Given u ∈ Sn−1 ⊂ R

n, let Tt : Rn+1 → R

n denote the (non-orthogonal when t 6= 0)projection onto R

n parallel to en+1 + tu. A family of convex bodies {K(t)}t∈R is calleda shadow system in the direction of u if there exists a compact convex set K ⊂ R

n+1 sothat K(t) = Tt(K).

Proposition 3.4. Let {K(t)}t∈R denote a shadow system in the direction of u ∈ Sn−1,

and let E ∈ Gu⊥,k−1. Then for any fixed s ∈ R, the function:

u⊥ ×R ∋ (y, t) 7→ |PE∧(y+su)K(t)|

is jointly convex in (y, t).

We will apply the above to a specific linear shadow system constructed from K andRuK, where recall Ru denotes reflection about u⊥. It was observed by Shephard in[59] that there exists a shadow system in the direction of u so that K(1) = K andK(−1) = RuK. Moreover, there exists a maximal shadow system {Ku(t)} with thisproperty, in the sense that Ku(t) ⊇ K(t) for any {K(t)} as above; indeed, it is givenby setting K := T−1

1 (K) ∩ T−1−1 (RuK). Equivalently, this maximal system is obtained

by replacing each one-dimensional section K(y) of K in the direction of u over y ∈ u⊥,

10

by the Minkowski sum 1+t2 K(y) + 1−t

2 RuK(y). Consequently, {Ku(t)}t∈[−1,1] constitutes

a Minkowski linear system [8], and in particular, |Ku(t)| = |K| for all t ∈ [−1, 1] andKu(0) = SuK. Note that RuKu(t) = Ku(−t) for all t ∈ R.

The proof of Proposition 3.4 is deferred to Subsection 3.3; an alternative proof ispresented in Section 4.

3.1.3 Blaschke–Petkantschin-type formula

The final crucial ingredient, without which we do not know how to obtain sharp lowerbounds on Qk,p(K) for p < −(k + 1) (see Remark 7.7), is the following Blaschke–Petkantschin-type formula. It is a particular case of [58, Theorem 7.2.6] (applied withq = 1, m = s1 = k, s0 = d− 1) from the excellent monograph of Scheinder–Weil.

Theorem 3.5. Fix u ∈ Sn−1. Then for any measurable function f : Gn,k → R+:

cn,k

∫

Gn,k

f(F )σn,k(dF ) =

∫

Gu⊥,k−1

∫

Sn−k(E⊥)f(span(E, θk)) |〈θk, u〉|k−1 dθkσu⊥,k−1(dE),

where σu⊥,k−1 is the uniform Haar probability measure on Gu⊥,k−1.

Here cn,k is an explicit positive constant depending only on n, k, whose value isimmaterial for us (it may be found in [58, Theorem 7.2.6]).

3.2 Proof of Theorem 3.1

Let k = 1, . . . , n − 1 be fixed. Given u ∈ Sn−1, introduce the following measure:

µu := |〈xk, u〉|k−1LE⊥(dxk)σu⊥,k−1(dE).

Lemma 3.6. For any compact set K ⊂ Rn and u ∈ S

n−1:

µu(Lk,u(K)) =cn,kn

∫

Gn,k

1

|PFK|nσn,k(dF ).

Proof. Set p(xk) := |〈xk, u〉|k−1. Integrating in polar coordinates on E⊥ and invokingTheorem 3.5, we obtain:

µu(Lk,u(K)) =

∫

Gu⊥,k−1

∫

E⊥

1Lk,u(K)(E, xk)p(xk)LE⊥(dxk)σu⊥,k−1(dE)

=

∫

Gu⊥,k−1

∫

Sn−k(E⊥)

∫ ∞

01Lk,u(K)(E, rθk)p(rθk)r

n−kdr dθk σu⊥,k−1(dE)

=

∫

Gu⊥,k−1

∫

Sn−k(E⊥)p(θk)

∫ 1/|Pspan(E,xk)(K)|

0rn−1dr dθk σu⊥,k−1(dE)

=1

n

∫

Gu⊥,k−1

∫

Sn−k(E⊥)

1

|Pspan(E,θk)K|n |〈θk, u〉|k−1 dθk σu⊥,k−1(dE)

=cn,kn

∫

Gn,k

1

|PFK|nσn,k(dF ).

11

Proof of Theorem 3.1. In view of Lemma 3.6, we would like to show that for any convexbody K:

µu(Lk,u(K)) ≤ µu(Lk,u(SuK)). (3.2)

The advantage of the latter formulation is that now everything is “aligned” with u, thedirection in which we perform the Steiner symmetrization. Consequently, we evaluatethings by decomposing each E⊥ into span(u)⊕ (E⊥ ∩ u⊥) and applying Fubini:

µu(Lk,u(K)) =

∫

Gu⊥,k−1

∫

E⊥

1LK(E, xk) |〈xk, u〉|k−1LE⊥(dxk)σu⊥,k−1(dE)

=

∫

Gu⊥,k−1

∫

R

∫

E⊥∩u⊥

1|PE∧(y+su)K|≤1 |〈y + su, u〉|k−1 dy ds σu⊥,k−1(dE)

=

∫

Gu⊥,k−1

∫

R

|s|k−1

∫

E⊥∩u⊥

1|PE∧(y+su)K|≤1dy ds σu⊥,k−1(dE)

=

∫

Gu⊥,k−1

∫

R

|s|k−1|LE,u,s(K)| ds σu⊥,k−1(dE), (3.3)

where we denote:

LE,u,s(A) := {y ∈ E⊥ ∩ u⊥ ; |PE∧(y+su)A| ≤ 1}.So far we haven’t used the convexity of K. We now apply the key Proposition 3.4 to

the linear shadow system Ku(t) from Subsection 3.1.2. As E ⊂ u⊥ and y ∈ E⊥ ∩ u⊥, itfollows that for every fixed s ∈ R, the function:

(E⊥ ∩ u⊥)× R ∋ (y, t) 7→ f (s)(y, t) := |PE∧(y+su)Ku(t)| is jointly convex.

In addition, f (s)(y, t) is an even function, since:

f (s)(−y,−t) = |PE∧(−y+su)Ku(−t)| = |PRuE∧Ru(−y+su)Ku(t)|= |PE∧(−y−su)Ku(t)| = |PE∧(y+su)Ku(t)| = f (s)(y, t).

Hence, its level set:

LE,u,s := {(y, t) ∈ (E⊥ ∩ u⊥)× R ; |PE∧(y+su)Ku(t)| ≤ 1}is an origin-symmetric convex body. Note that its t-section is precisely LE,u,s(Ku(t)).Inspecting the t-sections at t = −1, 0, 1 and recalling that Ku(1) = K and Ku(0) = SuK,convexity and origin-symmetry of LE,u,s imply:

LE,u,s(SuK) ⊇ 1

2(LE,u,s(K)− LE,u,s(K)). (3.4)

By the Brunn–Minkowski inequality, we deduce:

|LE,u,s(SuK)| ≥ |LE,u,s(K)|.Plugging this back into (3.3) and rolling everything back, we deduce the desired (3.2),thereby concluding the proof.

12

In fact, the above proof gives us more information:

Theorem 3.7. For any convex body K in Rn and u ∈ S

n−1, the function R+ ∋ t 7→Φk(Ku(t)) = Φk(Ku(−t)) is monotone non-decreasing.

Proof. As dim(E⊥ ∩ u⊥) = n − k, we actually know by Brunn’s concavity principle,

applied to the t-sections of LE,u,s, that the function R ∋ t 7→ |LE,u,s(Ku(t))|1

n−k isconcave on its support. It is also even by origin-symmetry of LE,u,s. In particular,

R+ ∋ t 7→ |LE,u,s(Ku(t))| = |LE,u,s(Ku(−t))| is non-increasing. (3.5)

Integrating this according to (3.3) and applying Lemma 3.6, the assertion follows.

3.3 Proof of convexity of shadow system projections

To complete the proof, it remains to establish Proposition 3.4.

Denote:|Px1∧...∧xk

K| := Lk(Pspan{x1,...,xk}K)∆(x1, . . . , xk),

where ∆(x1, . . . , xk) denotes the Lk measure of the parallelepiped [0, x1] + . . . + [0, xk].This is consistent with our previous notation introduced in (3.1) since if E ∈ Gn,k−1 isspanned by an orthonormal basis {x1, . . . , xk−1}, we clearly have:

|PE∧xkK| = |Px1∧...∧xk−1∧xk

K|.We remark that the method described in this subsection is quite general, and may

be used to show the following much more general version of Proposition 3.4:

Proposition 3.8. Let K(t) be a shadow system in the direction of u. Then for anyx1, . . . , xk ∈ u⊥, for any a1, . . . , ak ∈ R, and for any s ∈ R, the function:

(y, t) ∋ u⊥ × R 7→ |P(a1(su+y)+x1)∧...∧(ak(su+y)+xk)K(t)|is jointly convex.

However, when all of the ai’s are zero except for one, as in Proposition 3.4, a muchsimpler proof is available, and so we leave the verification of Proposition 3.8 to theinterested reader and focus on the former simple scenario.

Let us introduce some useful notation which we will frequently use throughout theanalysis of equality later on. Given w ∈ E, denote:

Kw := (K − w) ∩ E⊥,

and note that if x ∈ E⊥ then:

Pspan(E,x)K =⋃

w∈E

(w + Pspan(x)Kw).

Hence by Fubini and homogeneity, for all x ∈ E⊥:

|PE∧xK| =∫

E|PxK

w|dw =

∫

E(hKw(x) + hKw(−x))dw. (3.6)

13

Lemma 3.9. Let K be a convex compact set in Rn. For any linear subspace E, Rn ∋

x 7→ |PE∧xK| is convex. In particular, its level set LE(K) in E⊥ is convex.

Proof. Since |PE∧xK| only depends on PE⊥x, it is enough to establish convexity forx ∈ E⊥. But this is immediate from (3.6) and the convexity of the support functionshKw .

Note that (3.6) yields a useful expression for ‖x‖LE(K).

Remark 3.10. It is possible to prove the following more general claim: for any x1, . . . , xk ∈Rn and a1, . . . , ak ∈ R, the function R

n ∋ z 7→ |P(x1+a1z)∧...∧(xk+akz)K| is convex; weleave this to the interested reader.

The following linear-algebra lemma is elementary:

Lemma 3.11. For any linear map T : Rn → Rn and compact set A ⊂ R

n:

|Px1∧...∧xkT (A)| = |PT ∗(x1)∧...∧T ∗(xk)A|. (3.7)

We will only require to know that |PE∧xT (K)| = |PE∧T ∗(x)K| when T acts as the identity

on E and invariantly on E⊥, which is totally elementary and may be proved as in theprevious lemma by using that hT (Kw)(x) = hKw(T ∗x); however, for completeness, weprovide a proof of the general version above. First observe:

Lemma 3.12. For any linear map T : Rn → Rn and subspace E ⊂ Rn so that T ∗|E :E → T ∗E is injective, there exists a linear map S : T ∗E → E so that:

PE ◦ T = S ◦ PT ∗E . (3.8)

Proof. The operator M = T ∗ ◦ PE ◦ T is self-adjoint so ImM ⊆ T ∗E is an invariantsubspace of M . Since KerM = (ImM)⊥ ⊇ (T ∗E)⊥, we may therefore write:

T ∗ ◦ PE ◦ T = M = N ◦ PT ∗E ,

for some self-adjoint linear map N : T ∗E → T ∗E. It follows that (3.8) holds withS = (T ∗|E)−1 ◦N .

Proof of Lemma 3.11. We may assume that {xi} are linearly independent, otherwiseboth sides of (3.7) are zero and there is nothing to prove. Denote E = span{x1, . . . , xk}.Note that PE ◦ T is onto E iff (ImT )⊥ ∩ E = {0} iff KerT ∗ ∩ E = {0}, and so we mayassume that T ∗|E is injective, otherwise both sides of (3.7) are again zero. By Lemma3.12:

|PE ◦ T (A)| = |S ◦ PT ∗E(A)| = | detT ∗E→E

S||PT ∗E(A)|,

where |detP→QL|, the (constant) Jacobian of the linear map L : P → Q, is equal to√detP→P L∗L =

√

detQ→QLL∗. Employing (3.8):

| detT ∗E→E

S| =√

detE→E

SS∗ =√

detE→E

PETT ∗PE = | detE→T ∗E

T ∗|E |.

14

Hence:

|Px1∧...∧xkT (A)| = ∆(x1, . . . , xk)|PET (A)| = ∆(x1, . . . , xk)| det

E→T ∗ET ∗|E ||PT ∗E(A)|

= ∆(T ∗x1, . . . , T∗xk)|PT ∗E(A)| = |PT ∗(x1)∧...∧T ∗(xk)A|.

We are now ready to prove Proposition 3.4

Proof of Proposition 3.4. We know that K(t) = Tt(K) for some convex compact setK ⊂ R

n+1, where Tt is a projection which acts as identity on Rn and sends en+1 to

−tu. One immediately checks that T ∗t acts as the identity on u⊥ (and in particular on

E ⊂ u⊥) and T ∗t (u) = u− ten+1. Hence by Lemma 3.11, if y ∈ u⊥:

|PE∧(su+y)K(t)| = |PE∧(su−sten+1+y)K|.

(see also [16, (5)] for the case E = {0}). The convexity in (y, t) now follows from Lemma3.9, as the map (t, y) 7→ su− sten+1 + y is affine (for fixed s ∈ R).

3.4 Proof of the isoperimetric inequality

The conclusion of the proof of the inequality of Theorem 1.1 is now standard. It iswell known that given a compact set K ⊂ R

n, there exists a sequence of directions{ui}i=1,2,... ⊂ S

n−1 so that the compact sets Ki := SuiSui−1 . . . Su1K converge in the

Hausdorff topology to BK , the Euclidean ball having the same volume as K (see e.g.[27, Theorem 9.1] for the case that K is convex or [13, Lemma 9.4.3] for the generalcase). When K is in addition a convex body, since Steiner symmetrization preservesconvexity, all the Ki are convex bodies as well. Clearly Φk is continuous on the class ofconvex bodies with respect to the Hausdorff topology (e.g. [57, Theorem 1.8.20]), andhence by Theorem 3.1:

Φk(K) ≥ Φk(K1) ≥ . . . ≥ Φk(Ki) ց Φk(BK).

4 Further convexity properties

After having proved Proposition 3.4 (in fact, the more general Proposition 3.8), we ob-served that we may actually obtain a different proof of Proposition 3.4 which is modeledafter the Meyer–Pajor proof of the Blaschke–Santalo inequality from [44]. This proof hasthe advantage that it may be written so as to avoid any reference to shadow systems.Moreover, it highlights an intimate relation between Theorem 1.1 for general k and theparticular case k = 1 corresponding to the Blaschke–Santalo inequality, which in a senseunderlies our proof. Most importantly, it reveals a certain additional convexity propertyof |PE∧(y+su)K(t)| in s when s is varied harmonically (as in [45] for k = 1), which willbe crucially used in the characterization of local minimizers of Φk. On the other hand,a proof of the more general Proposition 3.8 seems to be out of reach of this approach.

15

4.1 Alternative proof of Proposition 3.4

To avoid any reference to shadow systems, we will only verify the convexity of |PE∧(y+su)K(t)|between the three sections at t = 1, 0,−1 and for K(t) = Ku(t), which is the only thingwe require for the proof of Theorem 3.1.

Proposition 4.1. Let K be a convex body in Rn, let u ∈ S

n−1 and E ∈ Gu⊥,k−1, and

fix s ∈ R. Then for all y1, y2 ∈ E⊥ ∩ u⊥:

|PE∧(

y1+y22

+su)SuK| ≤

|PE∧(y1+su)K|+ |PE∧(y2−su)K|2

.

Observe that this immediately implies the inequality (3.4) used in the proof of The-orem 3.1 (after noting that |PE∧(y2−su)K| = |PE∧(−y2+su)K|).

Proof. Recall the notation Kw = (K −w) ∩E⊥ for w ∈ E, and observe that (SuK)w =Su(K

w) since u ∈ E⊥. Recalling (3.6), it follows that it is enough to verify:

|P y1+y22

+suSuK

w| ≤ |Py1+suKw|+ |Py2−suK

w|2

for all w ∈ E. This is particularly convenient since all projections are one-dimensionalintervals. Parametrizing E⊥ as {(a, b) := a+ bu ; a ∈ E⊥ ∩ u⊥, b ∈ R}, we verify this asfollows:

|P y1+y22

+suSuK

w| = max{〈y1 + y22

, a1 − a2〉+ s(b1 − b2) ; (ai, bi) ∈ SuKw}

= max{〈y1 + y22

, a1 − a2〉+ s

(

r+1 − r−12

− r+2 − r−22

)

; (ai, r±i ) ∈ Kw} (4.1)

≤ 1

2max{〈y1, a1 − a2〉+ s(r+1 − r+2 ) ; (ai, r

+i ) ∈ Kw}

+1

2max{〈y2, a1 − a2〉 − s(r−1 − r−2 ) ; (ai, r

−i ) ∈ Kw}

=|Py1+suK

w|+ |Py2−suKw|

2.

Remark 4.2. For a general shadow system {K(t)} in the direction of u, if we denoteby K(b)(t) the one-dimensional section of K(t) over b ∈ u⊥ parallel to u, we necessarilyhave for all b ∈ u⊥:

K(b)

(

t1 + t22

)

⊆ 1

2K(b)(t1) +

1

2K(b)(t2) ∀t1, t2 ∈ R ;

for instance, one can see this by maximality of the shadow system K ′(t) := Tt(K′)

for K ′ := T−1t1 (K(t1)) ∩ T−1

t2 (K(t2)) (cf. [54] or [45, (4)]). So exactly the same proof asabove applies to a general shadow system, replacing the equality in (4.1) by an inequality.Repeating the proof for general values of t1, t2 ∈ R with their mid-point t1+t2

2 (in placeof 1,−1, 0 as above), one obtains an alternative proof of Proposition 3.4.

16

4.2 Harmonic convexity in s

Repeating verbatim the above proof and allowing the parameter s to vary, one obtainsthe following additional harmonic convexity in s. Such a property was first observed byMeyer–Reisner [45] for sections of the polar body (in fact, with respect to the Santalopoint), corresponding to the case k = 1.

Proposition 4.3. Let {K(t)} be a shadow system in the direction of u ∈ Sn−1. Let

E ∈ Gu⊥,k−1, and s1, s2 > 0. Then for all y1, y2 ∈ E⊥ ∩ u⊥ and t1, t2 ∈ R:

|PE∧(

s2y1+s1y2s1+s2

+2s1s2s1+s2

u)K

(

t1 + t22

)

| ≤ s2s1 + s2

|PE∧(y1+s1u)K(t1)|

+s1

s1 + s2|PE∧(y2+s2u)K(t2)|.

Consequently:

LE,u,

2s1s2s1+s2

(K

(

t1 + t22

)

) ⊇ s2s1 + s2

LE,u,s1(K(t1)) +s1

s1 + s2LE,u,s2(K(t2)),

and hence by the Brunn–Minkowski inequality:

|LE,u,

2s1s2s1+s2

(K

(

t1 + t22

)

)| ≥ |LE,u,s1(K(t1))|s2

s1+s2 |LE,u,s2(K(t2))|s1

s1+s2 . (4.2)

We now proceed as in [45], and invoke the following harmonic Prekopa–Leindler-typeinequality of K. Ball [3, p. 74] (see also [2, Theorems 1.4.6 and 10.2.10]):

Theorem 4.4 (Ball). Let f, g, h : R+ → R+ be measurable functions so that for alls1, s2 ∈ R+:

h

(

2s1s2s1 + s2

)

≥ f(s1)s2

s1+s2 g(s2)s1

s1+s2 .

Then for all p > 0, denoting Ip(w) :=(∫∞

0 sp−1w(s)ds)−1/p

, we have:

Ip(h) ≤1

2(Ip(f) + Ip(g)).

Corollary 4.5. With the same assumptions as in Proposition 4.3, the function

R ∋ t 7→(∫

LE(K(t))〈x, u〉k−1

+ dx

)−1/k

=

(∫ ∞

0sk−1|LE,u,s(K(t))|ds

)−1/k

is convex.

Proof. The convexity at the mid-point t1+t22 of t1, t2 ∈ R follows by an application

of Theorem 4.4 to (4.2). The general case follows by continuity (or by an obviousmodification of the above to general positive coefficients α+ β = 1).

In the case k = 1 and when {K(t)} are all origin symmetric, Corollary 4.5 amountsto the convexity of t 7→ |K(t)◦|−1, first established by Campi–Gronchi [16], and extendedby Meyer–Reisner to the convexity of t 7→ |K(t)◦,s|−1 for general convex bodies [45].

17

4.3 A dichotomy for t 7→ Φk(Ku(t))

Applying Corollary 4.5 to the linear shadow system Ku(t), for which

LE,u,s(Ku(−t)) = LE,u,−s(Ku(t)) = −LE,u,s(Ku(t)) = −LE,u,−s(Ku(−t)),

we deduce the convexity of the following function, appearing in (3.3):

Mk(LE(Ku(t))) :=

(

∫

LE(Ku(t))|〈x, u〉|k−1 dx

)−1/k

=

(∫

R

|s|k−1|LE,u,s(Ku(t))|ds)−1/k

.

Theorem 4.6. The function R ∋ t 7→ Mk(LE(Ku(t))) is convex and even.

By Theorem 3.7 we already know that R+ ∋ t 7→ Φk(Ku(t)) = Φk(Ku(−t)) ismonotone non-decreasing. The next theorem, in which the above convexity will becrucially used, adds vital information – this function transitions from being constant on[0, a] to strictly monotone on [a,∞) at a unique a ∈ [0,∞].

Theorem 4.7. Given u ∈ Sn−1 and t0 ∈ R, the equality Φk(Ku(t1)) = Φk(Ku(t0)) holds

for some |t1| < |t0| if and only if it holds for all |t1| < |t0|.

Proof. Assume Φk(Ku(t1)) = Φk(Ku(t0)) for some |t1| < |t0|, or equivalently (by Lemma3.6), µu(Lk,u(Ku(t1))) = µu(Lk,u(Ku(t0))). Recall from (3.3) that:

µu(Lk,u(Ku(t))) =

∫

Gu⊥,k−1

Mk(LE(Ku(t)))−kσu⊥,k−1(dE). (4.3)

Theorem 4.6 implies in particular that R+ ∋ t 7→ Mk(LE(Ku(t))) = Mk(LE(Ku(−t)))is monotone non-decreasing for all E ∈ Gu⊥,k−1 (alternatively, a simpler way to deducethe monotonicity is by (3.5)). It follows that necessarily:

Mk(LE(Ku(±t1))) = Mk(LE(Ku(±t0))),

for almost all E ∈ Gu⊥,k−1, and hence by continuity, for all E. Invoking now the fullstrength of Theorem 4.6, it follows that [−t0, t0] ∋ t 7→ Mk(LE(Ku(t))) must be constantfor all E. Recalling (4.3) and Lemma 3.6, we deduce that Φk(Ku(t)) = Φk(Ku(t0)) forall t ∈ [−t0, t0] .

Remark 4.8. One might try to prove Theorem 4.7 by expanding on the argument of

Theorem 3.7. By Brunn’s concavity principle, we know that R ∋ t 7→ |LE,u,s(Ku(t))|1

n−k

is concave and even on its support, but the problem is that for a given s ∈ R, the supportmay be a strict subset of [−t1, t1], and so we cannot conclude that the latter function isconstant on any interval.

The usefulness of a statement like Theorem 4.7 for characterizing local extremizerswas observed in the case k = 1 by Meyer–Reisner [46].

18

5 Analysis of equality

Let K be a convex body in Rn and fix k ∈ {1, . . . , n − 1}. In the next two sections we

will establish the equality case of Theorem 1.2:

Theorem 5.1. Φk(K) = Φk(SuK) for all u ∈ Sn−1 if and only if K is an ellipsoid.

Corollary 5.2. Φk(K) = Φk(BK) if and only if K is an ellipsoid.

Proof of Corollary 5.2 given Theorem 5.1. If K is an ellipsoid, the invariance of Φk un-der volume preserving affine maps implies that Φk(K) = Φk(BK). Conversely, since wealways have:

Φk(K) ≥ Φk(SuK) ≥ Φk(BK) ∀u ∈ Sn−1

by Theorems 3.1 and (1.2), if Φk(K) = Φk(BK) then we have equality above, and so Kmust be an ellipsoid by Theorem 5.1.

Proof of Theorem 1.3 given Theorem 5.1. Assume that K is a local minimizer of Φk

among all convex bodies of a given volume with respect to the Hausdorff topology.For every u ∈ S

n−1, since t 7→ Ku(t) is clearly continuous in the this topology, we knowthat there exists ǫ ∈ (0, 1) so that Φk(Ku(1 − ǫ)) ≥ Φk(K) (as |Ku(t)| = |K| for allt ∈ [−1, 1]). On the other hand, by Theorem 3.7, we know that Φk(K) ≥ Φk(Ku(t))for all t ∈ [−1, 1], and hence we must have equality at t = 1 − ǫ. Therefore, Theorem4.7 implies that we have equality for all t ∈ [−1, 1], and in particular at t = 0, i.e.Φk(K) = Φk(SuK). Since this holds for all u ∈ S

n−1, Theorem 5.1 implies that K mustbe an ellipsoid.

The trivial direction of Theorem 5.1 follows from the well-known fact (e.g. [9, Lemma2]) that Steiner symmetrization transforms an ellipsoid into an ellipsoid (of the samevolume), together with the affine-invariance of Φk. The proof of the non-trivial directionconsists of several steps. Steps 1 and 2 are inspired by the Meyer–Pajor simplification[44] of Saint-Raymond’s analysis in [55] of the equality case in the Blaschke–Santaloinequality for origin-symmetric convex bodies; however, to treat general convex bodies,we put forward several new observations in Steps 3 and 4 which are new even in theclassical case k = 1. In Step 5 we conclude the proof in the range 1 ≤ k ≤ n − 2. Theremaining case k = n− 1 requires more work, which is deferred to the next section.

5.1 Step 1 - point of symmetry

Let us recall several definitions introduced in the proof of Theorem 3.1. Given u ∈ Sn−1

and E ∈ Gu⊥,k−1, recall that:

LE(K) = {x ∈ E⊥ ; |PE∧xK| ≤ 1},

and that LE(K) is origin symmetric and convex by Lemma 3.9. Also recall that:

LE,u,s(K) = {y ∈ E⊥ ∩ u⊥ ; |PE∧(y+su)K| ≤ 1},

19

and note that LE,u,s(K) is precisely the section of LE(K) perpendicular to u at heights:

LE,u,s(K) = (LE(K)− su) ∩ u⊥. (5.1)

In particular, LE,u,s(K) is convex. Moreover, we know by (3.4) that:

LE,u,s(SuK) ⊇ 1

2(LE,u,s(K)− LE,u,s(K)), (5.2)

and hence by the Brunn–Minkowski inequality:

|LE,u,s(SuK)| ≥ |LE,u,s(K)|. (5.3)

Given u ∈ Sn−1, assume that Φk(K) = Φk(SuK), or equivalently, µu(Lk,u(K)) =

µu(Lk,u(SuK)). In view of (5.3) and (3.3), we necessarily have:

|LE,u,s(SuK)| = |LE,u,s(K)| (5.4)

for almost all E ∈ Gu⊥,k−1 and s ∈ R, and hence by continuity for all E, s. If Φk(K) =

Φk(SuK) for all u ∈ Sn−1, it follows that (5.4) holds for all E ∈ Gn,k−1, u ∈ S(E⊥) and

s ∈ R. By the equality case of the Brunn-Minkowski inequality, we deduce from (5.2)and (5.4) that

LE,u,s(K) = LE,u,s(SuK) + αE,u,s, (5.5)

where αE,u,s ∈ E⊥∩u⊥ is some translation vector. Since LE,u,s(SuK) is origin symmetric(as RuSuK = SuK), it follows that LE,u,s(K) has a point of symmetry.

5.2 Step 2 - Brunn’s characterization

We now invoke the following characterization of ellipsoids, originating in Brunn’s 1889Habilitation [12]:

Theorem 5.3 (Brunn’s characterization). Let L be a convex body in Rq, q ≥ 3, and let

2 ≤ p ≤ q − 1. Then L is an ellipsoid iff every non-empty p-dimensional section of Lhas a point of symmetry.

Proof. The case when L is a regular convex body in R3 and p = 2 is due to Brunn

[12, Chapter IV] (see [60, Section 4]). By reverse induction on p, it is clear that it isenough to establish the case p = q − 1, for which we refer to [43, Theorem 2.12.13]. Infact, it was shown by Olovjanishnikov [48] (cf. [60, Theorem 4.3]) that it is enough torestrict to hyperplane sections which divide the volume of L in a given ratio λ 6= 1. A farreaching generalization was obtained by Aitchison–Petty–Rogers [1], who showed that itis enough consider all p-dimensional sections which pass through a fixed point x0 in theinterior of L which is not a point of symmetry of L, if it has one. We refer to the surveys[60, 51, 28] for additional extensions and characterizations of ellipsoids.

20

Fix E ∈ Gn,k−1. Recalling (5.1) and that every LE,u,s(K) has a point of symmetryfor all u ∈ S(E⊥) and s ∈ R, it follows that when dimE⊥ = n−k+1 ≥ 3, i.e. k ≤ n−2,LE(K) is necessarily an ellipsoid in E⊥. We proceed assuming this is the case, and defertreating the case k = n− 1 to the next section.

Note that we are still far from concluding that K is an ellipsoid even in the casek = 1 (when E = {0} and LE(K) = (K −K)◦), since we have only shown that K −Kis an ellipsoid, which does not mean that K itself is an ellipsoid (but rather an affineimage of a convex body of constant width).

5.3 Step 3 - distinguished orthonormal basis

By Lemma 3.11, for any compact set A ⊂ Rn and invertible linear map T :

LE(T (A)) = {x ∈ E⊥ ; |PE∧xT (A)| ≤ 1} = {x ∈ E⊥ ; |PE∧T ∗(x)A| ≤ 1} = T−∗(LE(A)).

Since LE(K) is an (origin-symmetric) ellipsoid in E⊥, we may find a positive-definitelinear map TE on R

n so that TE acts as the identity on E, and on E⊥, it maps theEuclidean ball BE⊥ onto LE(K). Denoting:

KE := TEK,

it follows that:LE(KE) = T−∗

E (LE(K)) = BE⊥ . (5.6)

Let {ui}i=1,...,n−k+1 denote an orthonormal basis of E⊥ consisting of eigenvectors ofTE . As TE acts diagonally in this basis, observe that the actions of Sui

and TE commute.Hence:

LE(SuiKE) = LE(Sui

TE(K)) = LE(TE(SuiK)) = T−∗

E (LE(SuiK)). (5.7)

Now recall by (5.5) and (5.1) that:

∀s ∈ R ∃αE,ui,s ∈ u⊥i (LE(K)− sui) ∩ u⊥i = (LE(SuiK)− sui) ∩ u⊥i + αE,ui,s.

Applying T−∗E to the last identity, using that it acts invariantly on span(ui) and u⊥i , and

recalling (5.6) and (5.7), we deduce:

(BE⊥−sui)∩u⊥i = (LE(SuiKE)−sui)∩u⊥i +T−∗

E (αE,ui,s) ∀s ∈ R ∀i = 1, . . . , n−k+1.

Since (LE(SuK)−su)∩u⊥ is origin symmetric in E⊥∩u⊥, and this does not change undera linear transformation, we know that (LE(Sui

KE)− sui) ∩ u⊥i is also origin symmetricin E⊥ ∩ u⊥i for all s ∈ R. Since (BE⊥ − sui)∩ u⊥i is origin symmetric as well, we deducethat T−∗

E (αE,ui,s) = 0 necessarily. It follows that:

LE(SuiKE) = LE(KE) = BE⊥ ∀i = 1, . . . , n− k + 1. (5.8)

21

5.4 Step 4 - invariance under reflections

Lemma 5.4. Let K be a convex body in Rn, let E ∈ Gn,k−1 (k = 1, . . . , n − 1), and

denote Kw := (K − w) ∩ E⊥ for w ∈ E. Assume that LE(SuK) = LE(K) for someu ∈ E⊥. Then for every w ∈ E, up to translation in the direction of u, it holds thatSuK

w = Kw, i.e. Kw is invariant under Ru, the reflection about u⊥.

Proof. Given x ∈ E⊥, recall from (3.6) that:

‖x‖LE(K) = |PE∧xK| =∫

E|PxK

w|dw =

∫

E(hKw(x) + hKw(−x))dw.

We are given that LE(SuK) = LE(K), and since (SuK)w = SuKw for all w ∈ E (as

u ∈ E⊥), we deduce that:∫

E(hKw(x) + hKw(−x))dw =

∫

E(hSuKw(x) + hSuKw(−x))dw.

Note that SuKw ⊆ 1

2(Kw +RuK

w), and hence:

hSuKw ≤ 1

2(hKw + hRuKw).

Since hRuKw(ξ) = hKw(Ruξ), it follows that:∫

E(hKw(ξ) + hKw(−ξ))dw ≤ 1

2

∫

E(hKw(ξ) + hKw(Ruξ) + hKw(−ξ) + hKw(−Ruξ))dw.

Applying this to ξ = θ and ξ = Ruθ for a given θ ∈ E⊥, and summing, we obtain:∫

E(hKw(θ) + hKw(−θ) + hKw(Ruθ) + hKw(−Ruθ))dw

≤∫

E(hKw(θ) + hKw(Ruθ) + hKw(−θ) + hKw(−Ruθ))dw

Since both sides are equal, this means that we must have equality for a.e. w (andhence, by continuity of the corresponding functions on their support, for all w), in the 4inequalities we used above, and we deduce:

hSuKw(ξ) =1

2(hKw(ξ) + hRuKw(ξ)) ∀ξ ∈ {θ,−θ,Ruθ,−Ruθ}.

Since θ was arbitrary, it follows that for all w ∈ E:

SuKw =

1

2(Kw +RuK

w).

But by the Brunn–Minkowski inequality:

|Kw| = |SuKw| ≥ |Kw| 12 |RuK

w| 12 = |Kw|,

and the equality case implies that RuKw and Kw are translates. Since there cannot be

any translation perpendicular to u, the proof is concluded.

22

Fix w ∈ E. The lemma and (5.8) imply that up to translating in the direction ofui, we have Rui

KwE = Kw

E . Since the ui’s are all orthogonal, it follows that there isa single translation of Kw

E so that RuiKw

E = KwE for all i = 1, . . . , n − k + 1. Since

the composition of all Rui’s is precisely −Id on E⊥, we deduce that Kw

E has a point ofsymmetry. Recalling that KE = TE(K) and that TE acts as the identity on E, it followsthat Kw has a point of symmetry.

5.5 Step 5 - concluding when 1 ≤ k ≤ n− 2

We have shown that for every E ∈ Gn,k−1, for every w ∈ E, the section Kw = (K−w)∩E⊥ has a point of symmetry. It follows by Brunn’s Theorem 5.3 that whenever n ≥ 3and dimE⊥ = n− k + 1 ≥ 2, i.e. k ≤ n− 1, K must be an ellipsoid.

All in all this establishes Theorem 5.1 when 1 ≤ k ≤ n − 2 (and hence n ≥ 3). Thecase when k = n− 1 will be handled in the next section.

6 Analysis of equality when k = n− 1

To establish the case k = n− 1 of Theorem 5.1, we cannot invoke Brunn’s Theorem 5.3in Step 2 of the previous section, since dimE⊥ = 2 for E ∈ Gn,k−1. In this section wedescribe a more complicated argument for bypassing Step 2 when k = n− 1.

6.1 Linear boundary segments

We will need the following two-dimensional observation (compare with [45, Lemma 8],which is insufficient for our purposes). Recall our notation:

L(K) = L{0}(K) = (K −K)◦.

Proposition 6.1. Let {K(t)}t∈R denote a shadow system of convex bodies in R2 in the

direction of e2. Given two non-empty open intervals S, T ⊂ R, assume that there existfunctions a,Ψ : S → R so that (a(s) + Ψ(s)t, s) ∈ ∂L(K(t)) for all s ∈ S and t ∈ T .Then there exist c+, c− ∈ R so that Ψ(s) = c+s+ − c−s− for all s ∈ S.

Proof. By definition, there exists a convex compact set K in R3 so that K(t) = Tt(K)

where Tt : R3 → R2 is a projection onto R

2 parallel to e3 + te2. As in the proof ofProposition 3.4, we have:

‖(y, s)‖L(K(t)) = hK(t)(y, s) + hK(t)(−y,−s)

= hK(y, s,−st) + hK(−y,−s, st) = ‖(y, s,−st)‖L(K) .

Our assumption then yields the following local parametrization of the surface ∂L(K):

F (s, t) := (a(s) + Ψ(s)t, s,−st) ∈ ∂L(K) ∀s ∈ S ∀t ∈ T.

23

The convexity of L(K) implies that its boundary may locally be represented by a con-vex function f , which is therefore Lipschitz and hence differentiable almost-everywhereby Rademacher’s theorem. Moreover, by Alexandrov’s theorem (e.g. [27, Chapter 2]),f is twice-differentiable (in Alexandrov’s sense) almost-everywhere. At points of firstdifferentiability, two linearly independent tangent vectors to the boundary are given by:

∂sF (s, t) = (a′(s) + Ψ′(s)t, 1,−t) , ∂tF (s, t) = (Ψ(s), 0,−s),

and so the normal to the boundary is in the direction:

N := (s,−sa′(s)− sΨ′(s)t+ tΨ(s),Ψ(s)).

At points of second differentiability, the surface has a second-order Taylor expansiongoverned by the second fundamental form:

II :=⟨

∂2sF,N/|N |

⟩

ds2 + 2 〈∂s∂tF,N/|N |〉 dtds+⟨

∂2t F,N/|N |

⟩

dt2.

Since ∂2t F ≡ 0 and 〈∂s∂tF,N〉 = sΨ′(s) − Ψ(s), we see that unless that latter term

vanishes, II will have strictly negative determinant, implying that the surface has asaddle at that point, contradicting convexity.

We now claim that the only (locally) Lipschitz function Ψ which solves sΨ′(s) −Ψ(s) = 0 for almost every s ∈ S is of the form Ψ(s) = c+s+ − c−s−. Indeed, denoteS+ and S− the open subsets of S where (the continuous) Ψ is positive and negative,respectively. On S+ we have (log Ψ)′(s) = (log s)′ and so by (local) absolute continuityof log Ψ we deduce that Ψ(s) = cis (ci 6= 0) on each connected component S+,i of S+;since Ψ must vanish at the end-points of each connected component which lie in S, thisimplies that there is at most one connected component in each of S ∩ R+ and S ∩ R−,and that its end-point in S must be at s = 0. An analogous statement holds on S−.This implies that Ψ must be of the asserted form.

6.2 Step 1 - segments of constant projections of K

Fix E ∈ Gn,k−1 and u ∈ S(E⊥). The argument of Step 1 from the previous section givesus a little more information than was stated there. Given s ∈ R, recall the definition off (s) from the proof of Theorem 3.1:

(E⊥ ∩ u⊥)× R ∋ (y, t) 7→ f (s)(y, t) := |PE∧(y+su)Ku(t)|.

We know that f (s) is convex and even in (y, t), and hence its level set:

LE,u,s := {(y, t) ∈ (E⊥ ∩ u⊥)× R ; f (s)(y, t) ≤ 1}

is convex and origin symmetric. Note that LE,u,s(t) = LE,u,s(Ku(t)), where we denote byA(t) the t-section of A. By Brunn’s concavity principle as in the proof of Theorem 4.7,

it follows that R ∋ t 7→ |LE,u,s(t)|1

n−k is even and concave on its support.

24

If Φk(K) = Φk(SuK), we know that |LE,u,s(1)| = |LE,u,s(−1)| = |LE,u,s(0)|, and so[−1, 1] ∋ t 7→ |LE,u,s(t)| must be constant. By the equality case of the Brunn-Minkowskiinequality, we deduce that LE,u,s ∩ {t ∈ [−1, 1]} must be a tilted cylinder over theorigin-symmetric base LE,u,s(0) = LE,u,s(SuK) ⊂ E⊥ ∩ u⊥. Hence:

LE,u,s(Ku(t)) = LE,u,s(SuK) + αE,u,st ∀t ∈ [−1, 1],

extending (5.5).Now, denote for R > 0:

LE,u,s,R := {(y, t) ∈ (E⊥ ∩ u⊥)× R ; f (s)(y, t) ≤ R}.

By homogeneity of f (s)(y, t) in (y, s) and a simple rescaling:

LE,u,s,R(t) = RLE,u,s/R,1(t) ∀t,

and hence:LE,u,s,R(t) = RLE,u,s/R(SuK) +RαE,u,s/Rt ∀t ∈ [−1, 1].

Using evenness, it follows that for every y ∈ E⊥ ∩ u⊥ so that f (s)(y, 0) = R,

f (s) ≡ R on both segments {(±y +RαE,u,s/Rt, t) ; t ∈ [−1, 1]}.

6.3 Step 2 - segments of constant projections of Kw

Given w ∈ E, recall that Kw = (K − w) ∩ E⊥. Note that (Kw)u(t) = (Ku(t))w, and

so we simply denote this by Kwu (t). Let W := {w ∈ E ; Kw 6= ∅}, and for w ∈ W and

s ∈ R denote:

(E⊥ ∩ u⊥)× R ∋ (y, t) 7→ f (s)w (y, t) := |Py+suK

wu (t)|,

so that:L(Kw

u (t)) = {y + su ; f (s)w (y, t) ≤ 1}.

Recall from (3.6) that:

f (s)(y, t) =

∫

Wf (s)w (y, t)dw, (6.1)

and that each f(s)w is convex and even in (y, t) by Proposition 3.4.

Denote by Σw(t) the compact interval Pspan(u)L(Kwu (t)) which we identify with a

subset of R (via su ↔ s). We claim that:

Σw := Σw(0) = Σw(t) ∀t ∈ [−1, 1].

Indeed, since the projection of the polar equals the polar of the section:

Pspan(u)L(Kwu (t)) = Pspan(u)(K

wu (t)−Kw

u (t))◦ = ((Kw

u (t)−Kwu (t)) ∩ span(u))◦.

25

But since Kwu (t) = ∪y∈P

u⊥Kw(y+(cw(y)t+[−ℓw(y), ℓw(y)])u) for all t ∈ [−1, 1], we have:

(Kwu (t)−Kw

u (t)) ∩ span(u) = ∪y∈Pu⊥

Kw [−2ℓw(y), 2ℓw(y)]u,

which is independent of t.

Now fix w0 ∈ W . Let±ys+su ∈ ∂L(Kw0u (0)) for s ∈ Σw0 , amounting to f

(s)w0 (±ys, 0) =

1. Denote Rys,s := f (s)(±ys, 0). Since W ∋ w 7→ f(s)w (x) is continuous for every

x = (y, t), since f(s)w are all convex, and since f (s) ≡ Rys,s on both segments {(±ys +

Rys,sαE,u,s/Rys,st, t) ; t ∈ [−1, 1]}, it follows that each f

(s)w must be constant on these

two segments as well, and in particular:

f (s)w0

≡ 1 on both segments {(±ys +Rys,sαE,u,s/Rys,st, t) ; t ∈ [−1, 1]}.

By convexity of f(s)w0 , this implies that for all s ∈ Σw0 and ±ys + su ∈ ∂L(Kw0

u (0)) wehave:

±ys +Rys,sαE,u,s/Rys,st+ su ∈ ∂L(Kw0

u (t)) ∀t ∈ [−1, 1].

6.4 Step 3 - using k = n− 1

When k = n−1 we have dimE⊥ = 2, and so theKw’s are two-dimensional convex bodies.Given A ⊂ E⊥, let us denote by A(s) the one-dimensional chord (A− su) ∩ (E⊥ ∩ u⊥),which we identify with a subset of R. The discussion in Step 2 implies that for all w ∈ W :

L(Kwu (t))(s) = [−aw(s), aw(s)] + Ψw(s)t ∀s ∈ Σw ∀t ∈ [−1, 1]. (6.2)

It follows by Proposition 6.1 that Ψw(s) = cw+s+ − cw−s− for some cw± ∈ R and alls ∈ Σw = Pspan(u)L(K

wu (t)) (the claim on the interior of Σw extends by continuity to

the entire Σw). But origin-symmetry of L(Kw) and the representation (6.2) for t = 1implies that Ψw must be odd, and hence cw := cw+ = cw−. We deduce that the mid-pointof the chord of L(Kw) perpendicular to u at height s is cws for all values of s for whichthe chord is non-empty, and hence all mid-points lie on a single line. This remains truefor any u ∈ S(E⊥), since we assume equality Φk(K) = Φk(SuK) for all directions u.

We can now invoke the following classical characterization of ellipsoids, due to Bertrand[5] and to Brunn [12, Chapter IV] (see the historical discussion in [60, Section 8] and [43,Theorem 2.12.1] or [23, Theorem 9.2.4] for a proof). Their original statement applied tothe plane, but easily extends to R

n:

Theorem 6.2 (Bertrand–Brunn). Let K be a convex body in Rn. Then K is an ellipsoid

if and only if for any direction u, the mid-points of all (one-dimensional) chords of Kparallel to u lie in a hyperplane.

We deduce from the Bertrand–Brunn Theorem that L(Kw) must be an (origin-symmetric) ellipsoid.

26

6.5 Step 4 - concluding the proof

We know that L(Kw) = Tw(BE⊥) for some linear map Tw : E⊥ → E⊥ and that

L(Kw)(s) = L(SuKw)(s) + cws ∀s.

We may therefore invoke the argument of Step 3 of the previous section to deduce thatthere exist two orthogonal directions u1, u2 ∈ S(E⊥) so that:

L(SuiTw(K

w)) = L(Tw(Kw)) = BE⊥ i = 1, 2.

Invoking the argument of Step 4 of the previous section, it follows that Tw(Kw) is

invariant (up to translation in the direction ui) under reflection about u⊥i , and henceTw(K

w) and therefore Kw have a point of symmetry. It follows as in Step 5 of theprevious section that if n ≥ 3 then K must be an ellipsoid by Brunn’s Theorem 5.3.

When n = 2 things are even simpler, since E = {0} and so K = Kw for w = 0; weknow that T0(K) = x0 + C for some origin-symmetric convex body C, and that:

(2C)◦ = (T0(K)− T0(K))◦ = L(T0(K)) = B22 ,

implying that C = 12B

22 and henceK is an ellipsoid. This concludes the proof of Theorem

5.1 when k = n− 1.

7 Lp-moment quermassintegrals and Alexandrov–Fenchel-

type inequalities

7.1 Lp-moment quermassintegrals

Definition 7.1. Given k = 1, . . . , n and p ∈ R, denote the Lp-moment quermassintegralsof a convex body K in R

n as:

Qk,p(K) :=|Bn

2 ||Bk

2 |

(

∫

Gn,k

|PFK|pσ(dF )

)1/p

.

The case p = 0 is interpreted in the limiting sense as:

Qk,0(K) =|Bn

2 ||Bk

2 |exp

(

∫

Gn,k

log |PFK|σ(dF )

)

.

Note that when p = −n we recover the affine quermassintegrals:

Qk,−n(K) = Φk(K).

It is known (see [26]) that p = −n is the unique value of p ∈ R for which Qk,p(K) isinvariant under volume-preserving affine transformations of K (explaining why we preferto use the notation Qk,p instead of Φk,p).

27

When p = 1, we obtain by Kubota’s formula [57, p. 301], [58, p. 222] the classicalquermassintegrals:

Qk,1(K) = Wk(K),

defined as the coefficients in Steiner’s formula:

|K + tBn2 | =

n∑

k=0

(

n

k

)

Wk(K)tn−k.

We continue our convention of using the index k instead of the more traditional n − kabove. The case p = −1 corresponds to the harmonic quermassintegrals Wk(K) definedby Hadwiger [29] and studied by Lutwak [33, 35]. We will present an interpretation ofthe case p = 0 as an averaged version of the Loomis-Whitney inequality in the nextsubsection.

Having Theorem 1.1 at hand, we can easily deduce:

Theorem 7.2. For any convex body K in Rn, k = 1, . . . , n − 1 and p > −n:

Qk,p(K) ≥ Qk,p(BK), (7.1)

with equality iff K is a Euclidean ball.

For p = 1, this is the classical isoperimetric inequality for intrinsic volumes [57, 27](e.g. the case k = n − 1 is the isoperimetric inequality for surface area, and the casek = 1 is Urysohn’s inequality for the mean width). For p = −1, the above isoperimetricinequality for the harmonic quermassintegrals was obtained by Lutwak in [35]. For−n < p < −1, this appears to be new.

Proof of Theorem 7.2. By Jensen’s inequality and Theorem 1.1:

Qk,p(K) ≥ Qk,−n(K) = Φk(K) ≥ Φk(Bk) = Qk,−n(BK) = Qk,p(BK).

If Qk,p(K) = Qk,p(BK) then we have equality in both inequalities above. Equality inthe second implies by Theorem 5.1 that K is an ellipsoid. Equality in the first (Jensen’sinequality) implies that Gn,k ∋ F 7→ |PFK| is constant, and henceK must be a Euclideanball.

Remark 7.3. The value of p = −n is precisely the sharp threshold below which theinequality (7.1) is no longer true, even for ellipsoids. Indeed, if p < −n and K is anyellipsoid which is not a Euclidean ball, then by Jensen’s inequality (which yields a strictinequality since Gn,k ∋ E 7→ |PEK| is not constant) and affine invariance of Qk,−n:

Qk,p(K) < Qk,−n(K) = Qk,−n(BK) = Qk,p(BK).

28

7.2 Case p = 0 - averaged Loomis–Whitney

The classical Loomis–Whitney inequality (for sets) [32] asserts that if K is a compactset in R

n then:Πn

i=1|Pe⊥iK| ≥ |K|n−1, (7.2)

with equality when K is a box (i.e. a rectangular parallelepiped with facets parallel tothe coordinate axes). From this, Loomis and Whitney deduce in [32] a non-sharp formof the isoperimetric inequality for the surface area S(K) (for any reasonable definitionof the latter):

S(K) ≥ 2|K|n−1n . (7.3)

Let Ik := {I ⊂ {1, . . . , n} ; |I| = k}, and given I ∈ Ik, denote EI = span{ei ; i ∈ I}.By reverse induction on k, it is easy to deduce the following extension of (7.2) to anyk = 1, . . . , n − 1:

ΠI∈Ik |PEIK| ≥ |K|(

n−1k−1),

with equality when K is a box. In the class of convex bodies, this is also a necessarycondition for equality. See [4, 62, 7] for further extensions.

Of course, one can choose any other orthonormal basis in the Loomis–Whitney in-equality instead of {e1, . . . , en}, and it is a natural question to ask whether a betterinequality holds if we average over all possible orthonormal bases (as a cube is not in-variant under rotations). Taking a geometric average gives a particularly pleasing result,since:

∫

SO(n)log ΠI∈Ik |PU(EI )K|σSO(n)(dU) =

(

n

k

)∫

Gn,k

log |PEK|σn,k(dE)

=

(

n

k

)

log

( |Bk2 |

|Bn2 |Qk,0(K)

)

.

Consequently, we have by Theorem 7.2 (in fact, this already follows from Lutwak’sconfirmation in [35] of the case p = −1), that for any convex body K in R

n and k =1, . . . , n− 1:

exp

(

∫

SO(n)log ΠI∈Ik |PU(EI )K|σSO(n)(dU)

)

≥( |Bk

2 ||Bn

2 |Qk,0(BK)

)(nk)

= |Bk2 |(

nk)( |K||Bn

2 |

)(n−1k−1)

,

with equality if and only if K is a Euclidean ball. This means that for the averagedLoomis–Whitney inequality, it is not the cube which is optimal but rather the Euclideanball. Moreover, since by Jensen’s inequality and Theorem 7.2:

1

nS(K) = Wn−1(K) = Qn−1,1(K) ≥ Qn−1,0(K) ≥ Qn−1,0(BK) = |Bn

2 |( |K||Bn

2 |

)n−1n

,

this averaged version implies the classical isoperimetric inequality (for convex bodies)with a sharp constant, in contrast to the non-sharp (7.3).

29

7.3 Alexandrov–Fenchel-type inequalities

It was noted by Lutwak in [33], following Hadwiger [29] for the case p = −1, that:

Q1/kk,p (K1 +K2) ≥ Q1/k

k,p (K1) +Q1/kk,p (K2) ∀kp ≤ 1.

In particular, this holds for all k = 1, . . . , n−1 when p ≤ 0. Indeed, this follows from theclassical Brunn–Minkowski inequality for PF (K1+K2) = PFK1+PFK2 and the reversetriangle inequality for the Lkp-norm when kp ≤ 1. A nice feature is that this extends toall compact sets K1,K2 (by Lusternik’s extension of the Brunn–Minkowski inequality tocompact sets [13, Section 8]).

This suggests that perhaps there is some Brunn–Minkowski-type theory for the Lp-moment quermassintegrals when p ≤ 0, and of particular interest is the affine-invariantcase p = −n.

It will be more convenient to use the following normalization, already defined in theIntroduction:

Ik,p(K) :=Qk,p(K)

Qk,p(BK)=

(∫

Gn,k|PFK|pσ(dF )

∫

Gn,k|PFBK |pσ(dF )

)1/p

.

Note that Ik,p(B) = 1 for any Euclidean ball B and all k, p, that In,p(K) = 1 for all p,that by Jensen’s inequality:

[−n, 1] ∋ p 7→ Ik,p(K) is non-decreasing,

and that Theorems 1.1 and 7.2 imply:

Ik,p(K) ≥ 1 ∀p ≥ −n, (7.4)

with equality for p > −n (p = −n) if and only if K is a Euclidean ball (ellipsoid).

In the classical case p = 1, Alexandrov’s inequalities [57, 22] (a particular case of theAlexandrov–Fenchel inequalities) assert that:

I1,1(K) ≥ I1/22,1 (K) ≥ . . . ≥ I1/k

k,1 (K) ≥ . . . ≥ I1/(n−1)n−1,1 (K) ≥ I1/n

n,1 (K) = 1.

In view of all of the above, the following was proved by Lutwak for p = −1 andconjectured to hold for p = −n in [35] (see also [23, Problem 9.5]):

Conjecture 7.4. For all p ∈ [−n, 0] and for any convex body K in Rn:

I1,p(K) ≥ I1/22,p (K) ≥ . . . ≥ I1/k

k,p (K) ≥ . . . ≥ I1/(n−1)n−1,p (K) ≥ I1/n

n,p (K) = 1.

Our isoperimetric inequality (7.4) establishes the inequality between each of the termsand the last one. Theorem 1.5 from the Introduction, which we repeat here for conve-nience, confirms “half” of the above conjecture.

30

Theorem 7.5. For every p ∈ [−n, 0] and 1 ≤ k ≤ m ≤ n:

I1/kk,p (K) ≥ I1/m

m,p (K),

for any convex body K in Rn whenever m ≥ −p. When k < m < n, equality holds for

p ≥ −m if and only if K is a Euclidean ball. When k < m = n, equality holds forp > −n (p = −n) if and only if K is a Euclidean ball (ellipsoid).

This confirms the conjecture for all p ∈ [−2, 0] and recovers the case p = −1 es-tablished by Lutwak in [35]. Our argument is very similar to the one used by Lutwak;however, instead of relying on Petty’s projection inequality, we have the full strength ofTheorem 1.1 at our disposal, which is crucial for handling the range p < −(k + 1).

As remarked in the introduction, the analogous statement for the dual Lp-momentquermassintegrals Ik,p was established by Gardner [24, Theorem 7.4] in exactly the samecorresponding range of parameters as in Theorem 7.5. Namely, defining

Ik,p(K) :=

(∫

Gn,k|K ∩ F |pσ(dF )

∫

Gn,k|BK ∩ F |pσ(dF )

)1/p

,

Gardner proved that I1/kk,p (K) ≤ I1/m

m,p (K) for an arbitrary bounded Borel set K andfor all 1 ≤ k ≤ m ≤ n and p ∈ (0,m], with precise characterization of the equalityconditions. However, he also showed in [24, Theorem 7.7] that an analogous statementto Conjecture 7.4 cannot hold for p = n, k = 1 and 2 ≤ m ≤ n−1: any origin-symmetricstar body K which is not an ellipsoid satisfies I1,n(K) = In,n(K) = 1, but Im,n(K) < 1.This troubling sign is likely a special feature of the dual setting when k = 1, sinceI1,n(K) = 1 for all origin-symmetric star bodies, whereas Im,n(K) = 1 if and only if Kis an origin-symmetric ellipsoid (up to a null-set) when 2 ≤ m ≤ n − 1 [24, Corollary7.5]. This abnormal behavior of the case k = 1 does not occur in the primal setting,thanks to the equality conditions of the Blaschke–Santalo inequality.

Proposition 7.6. For all 1 ≤ k ≤ m ≤ n and q ≤ m:

I1/kk,−q(K) ≥ I1/m

m,−q km

(K). (7.5)

When k < m, equality holds for q < m (q = m) if and only if K is a Euclidean ball(ellipsoid).

Proof. Applying (7.4) for the convex body PEK in the inner integral below, if 0 < q ≤m = dimE we have:

∫

Gn,k

|PFK|−qσn,k(dF ) =

∫

Gn,m

∫

GE,k

|PFPEK|−qσE,k(dF )σn,m(dE)

≤ c1

∫

Gn,m

|PEK|−q kmσn,m(dE). (7.6)

31

Taking the −q-th root, and then the k-th root, we obtain:

I1/kk,−q(K) ≥ c2I1/m

m,−q km

(K),

for some constants c1, c2 > 0 independent of K, for which equality holds when K is aEuclidean ball; it follows that necessarily c2 = 1. Similarly, if q < 0, we obtain thefirst inequality only reversed, and after taking the −q-th root, it remains in the correctdirection for us. The case q = 0 follows similarly.

Finally, if k < m and we have equality in (7.5), or equivalently in (7.6), it followsthat Ik,−q(PEK) = 1 for almost every E ∈ Gn,m, and hence for all E by continuity.Recalling the equality cases in (7.4), we deduce when q < m (q = m) that PEK is aEuclidean ball (ellipsoid) for all E ∈ Gn,m. If m = n this concludes the proof; otherwise,1 ≤ k < m ≤ n − 1, and it follows that K itself is a Euclidean ball (ellipsoid) by [23,Corollary 3.1.6 (Theorem 3.1.7)].

Proof of Theorem 7.5. For any 0 ≤ q ≤ m and 1 ≤ k ≤ m ≤ n, apply Proposition 7.6followed by Jensen’s inequality:

I1/kk,−q(K) ≥ I1/m

m,−q km

(K) ≥ I1/mm,−q(K).

Equality between the first and last terms implies equality throughout. If k < m, equalityin the first inequality occurs by Proposition 7.6 when q < m (q = m) if and only if Kis a Euclidean ball (ellipsoid). Equality in the second (Jensen’s) inequality occurs whenq > 0 if and only if Gn,m ∋ E 7→ |PEK| is constant, which for an ellipsoid K andm ≤ n− 1 implies that it must be a Euclidean ball; when q = 0 we cannot have q = mand so K is already known to be a ball.

Remark 7.7. Observe that when m = k+1, the above argument only relies on Petty’sprojection inequality. Consequently, Petty’s inequality in combination with Jensen’sinequality directly yield for any k = 2, . . . , n− 2:

I1/kk,−(k+1)(K) ≥ I1/(k+1)

k+1,−k (K) ≥ I1/(k+1)k+1,−(k+2)(K).

Continuing the chain of inequalities, one obtains:

I1/kk,−(k+1)(K) ≥ I1/(n−1)

n−1,−n (K),

which by a final application of Petty’s projection inequality is bounded below by 1. Weconclude (without relying on Theorem 1.1 and only using Petty’s inequality) that:

Ik,−(k+1)(K) ≥ 1 ∀k = 2, . . . , n− 2. (7.7)

It does not seem possible to improve the −(k + 1)-moment in (7.7) to the optimal one−n from Theorem 1.1 by using similar bootstrap arguments.

32

8 Concluding remarks

8.1 Extension to compact sets

In [23, Problem 9.4], Gardner asks whether it would be possible to extend Lutwak’sconjectured inequality of Theorem 1.1 to arbitrary compact sets. Certainly, our proof ofProposition 3.4 (in both Sections 3 and 4) employed convexity in an essential way, and itis not hard to show that the main claims there are simply false for general compact sets.However, the end result of Theorem 1.1 may very well be valid for general compact sets(as it is hard to imagine a non-convex set which would be more efficient than a Euclideanball). We briefly provide several remarks in this direction.

First, note that Lemma 3.9 only requires for each Kw to be connected. With a littlemore work, we can thus establish Theorem 3.1 for a fixed direction u ∈ S

n−1, for compactsets K so that for each E ∈ Gu⊥,k−1, every section of K parallel to E⊥ is connected.However, this property will be destroyed after applying Steiner symmetrization in aconsecutive sequence of directions, and so it is not clear how to exploit this to obtainthe end result of Theorem 1.1.

Second, observe that the inequality of Theorem 1.1 immediately extends to compactsets K whose k-dimensional projections are all convex. Indeed, simply apply Theo-rem 1.1 to conv(K), the convex-hull of K, which can only increase the volume of Kwhile preserving the volumes of all k-dimensional projections. Equality occurs if andonly if conv(K) is an ellipsoid and |K| = |conv(K)|. In particular, this already showsthat Theorem 1.1 for k = 1 remains valid for all connected compact sets K. However,we also observe that:

Theorem 8.1 (Blaschke–Santalo for compact sets). The inequality of Theorem 1.1 fork = 1 remains valid for arbitrary compact sets K, namely Φ1(K) ≥ Φ1(BK).

This provides an interpretation of the Blaschke–Santalo inequality which is transla-tion invariant, and which remains valid without any assumptions on convexity nor choiceof an appropriate center for K (see also [38, Corollary 6.4]).

Proof. Given the compact set K, define the following convex body:

K := ∩θ∈Sn−1 {x ∈ Rn ; |〈x, θ〉| ≤ |PθK| /2} .

We claim that:|K| ≤ |K|. (8.1)

To see this, recall that the Rogers–Brascamp–Lieb–Lutinger inequality [53, 11] assertsthat:

∫

Rn

Πmi=1fi(〈x, θi〉)dx ≤

∫

Rn

Πmi=1f

∗i (〈x, θi〉)dx,

for any measurable functions fi : R → R+ and directions θi ∈ Sn−1. Here f∗

i denote thesymmetric decreasing rearrangement of fi (see [11] for more details). Applying this to

33

fi := 1·θi∈PθiK(for which f∗

i = 1[−|PθiK|/2,|Pθi

K|/2]), we obtain:

|K| ≤ | ∩mi=1 {x ∈ R

n ; Pθix ∈ PθiK} | ≤ | ∩mi=1 {x ∈ R

n ; |〈x, θi〉| ≤ |PθiK| /2} |.

Using an increasing set of directions {θi}mi=1 which becomes dense in Sn−1, (8.1) easily

follows.On the other hand, we clearly have |PθK| ≤ |PθK| for all θ ∈ S

n−1, and so:

Φ1(K) ≥ Φ1(K) ≥ Φ1(BK) ≥ Φ1(BK). (8.2)

Remark 8.2. If there is equality between the left and right terms in (8.2), we musthave equality in all three inequalities. By Theorem 1.1, the second equality implies thatthe (origin-symmetric) convex body K is an ellipsoid. The third equality implies that|K| = |K|. Utilizing the first equality is non-trivial, see [18], and so we leave the analysisof equality for another occasion.

Remark 8.3. The above argument for k = 1 does not extend to general k > 1. Thereason is that the analogue of the Rogers–Brascamp–Lieb–Luttinger inequality for pro-jections onto dimension larger than one is false without some type of separability condi-tions on the projections (as in [11, Theorem 3.4]). This may be seen, for instance, by thesharpness of the Loomis–Whitney inequality (7.2) on cubes (as opposed to intersectionof spherical cylinders).

In summary, we presently do not know how to extend Theorem 1.1 for k > 1 togeneral compact sets. In particular, this applies to the classical case k = n − 1. Arelaxed variant of such a generalization is to replace the projection volume |PFK| by itsintegral-geometric version

∫

F χ(K∩(f+F⊥))df , where χ denotes the Euler characteristic;the two versions coincide when K is convex, but the latter may in general be larger thanthe former. For this relaxed variant, an extension of the case k = n− 1 to compact setsK with C1 smooth boundary was obtained by Zhang in [62].

Finally, we remark that the averaged Loomis–Whitney inequalityQk,0(K) ≥ Qk,0(BK)from Subsection 7.2 does extend to general compact sets K, but this requires a totallydifferent argument than the one presented here and will be verified elsewhere.

8.2 Simple new proof of Petty’s projection inequality

Our approach in this work suggests that the inequality Φk(K) ≥ Φk(BK) should beinterpreted as a generalization of the Blaschke–Santalo inequality, corresponding to thecase k = 1. It is a priori equally likely that it could be derived by generalizing Petty’sprojection inequality, corresponding to the other extremal case k = n − 1. In fact,we have spent a lot of time trying to derive it “from the Petty side”, without suc-cess. Our numerous attempts (see e.g. Remark 7.7) all ended up with the inequalityQk,−(k+1)(K) ≥ Qk,−(k+1)(BK), having the wrong power −(k+1) instead of the conjec-tured optimal −n. It would be interesting to give an alternative proof of Theorem 1.1for any k ∈ {1, . . . , n− 2} “from the Petty side”.

34

However, one useful byproduct of our failed attempts was the discovery of a newproof of Petty’s projection inequality, which is arguably the simplest proof we know.In particular, it completely avoids using the Busemann–Petty centroid inequality [23,Corollary 9.2.7]. Moreover, it seems to be a “dual version” of the Meyer–Pajor proof ofthe Blaschke–Santalo inequality [44]. We conclude this work by describing it.

Given a convex body K, recall the definition of the polar projection body Π∗K,whose associated norm is given by:

‖θ‖Π∗K = |Pθ⊥K| = nV (K,n − 1; [0, θ]) , θ ∈ Sn−1,

where V (K,n − 1;C) denotes the mixed volume of K (repeated n − 1 times) and C,and [0, x] denotes the segment between the origin and x (see e.g. [57, 23]). By homo-geneity, equality between the first and last terms above continues to hold for all θ ∈ R

n.Integration in polar-coordinates immediately verifies that:

|Π∗K| = 1

n

∫

Sn−1

|Pθ⊥K|−ndθ,

and so our goal is to show that:

|Π∗K| ≤ |Π∗SuK| (8.3)

(originally established by Lutwak–Yang–Zhang [37, 39]). We will in fact show a muchstronger claim, from which (8.3) immediately follows after integrating over u⊥:

Proposition 8.4. For all y ∈ u⊥, |(Π∗K)(y)| ≤ |(Π∗SuK)(y)|, where L(y) = {s ∈R ; y + su ∈ L} is the one-dimensional section of L parallel to u at y.

Proof. Fix y ∈ u⊥ and calculate:

|(Π∗K)(y)| =∫

R

1‖y+su‖Π∗K≤1ds =

∫

R

1V (K,n−1;[0,y+su])≤ 1nds. (8.4)

Consider the linear shadow system {Ku(t)} from Subsection 3.1.2. It easily follows fromShephard’s paper [59] that the function:

R2 ∋ (s, t) 7→ f(s, t) := V (Ku(t), n − 1; [0, y + su]) is jointly convex

(as the projections of Ku(t) and [0, y+ su] onto u⊥ do not depend on t, s). The functionf is also even since:

V (Ku(−t), n−1; [0, y−su]) = V (RuKu(t), n−1;Ru[0, y+su]) = V (Ku(t), n−1; [0, y+su]).

Hence its level set {(s, t) ∈ R2 ; V (Ku(t);n−1, [0, y+su]) ≤ 1/n} is an origin-symmetric

convex body, and so its section at t = 1 has smaller length than the one at t = 0:∫

R

1V (K,n−1;[0,y+su])≤ 1nds ≤

∫

R

1V (SuK,n−1;[0,y+su])≤ 1nds.

Plugging this into (8.4) and rolling back, the assertion follows.

35

Note that instead of fixing s (the u-height parameter) and integrating over y (per-pendicular to u) as in [44] and Section 3, we fix y and integrate over s. In all cases,the only inequality used in the proof is between two (n− k)-dimensional volumes (whichmay be thought of as the volumes of two t-sections of an (n− k+1)-dimensional convexbody).

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